Life can be defined has having:
Homeostasis: Regulation of the internal environment to maintain a constant state; for example, electrolyte concentration or sweating to reduce temperature.
I think hair regulates how oily it is, or dry.
Organization: Being structurally composed of one or more cells — the basic units of life.
Hair is organized by many smaller cells.
Metabolism: Transformation of energy by converting chemicals and energy into cellular components (anabolism) and decomposing organic matter (catabolism). Living things require energy to maintain internal organization (homeostasis) and to produce the other phenomena associated with life.
Growth: Maintenance of a higher rate of anabolism than catabolism. A growing organism increases in size in all of its parts, rather than simply accumulating matter.
Yes. Hair grows.
Adaptation: The ability to change over time in response to the environment. This ability is fundamental to the process of evolution and is determined by the organism's heredity, diet, and external factors.
Yes. Hair changes shape based on environment.
Response to stimuli: A response can take many forms, from the contraction of a unicellular organism to external chemicals, to complex reactions involving all the senses of multicellular organisms. A response is often expressed by motion; for example, the leaves of a plant turning toward the sun (phototropism), and chemotaxis.
Reproduction: The ability to produce new individual organisms,
Hair cannot reproduce, so I would say it is not alive if it had to meet every requirement, but these definitions are just guidelines and many things are alive that only meet some of them.